Waveform relaxation of nonlinear second-order differential equations

In this paper, we give a simple theorem on the waveform relaxation (WR) solution for a system of nonlinear second-order differential equations. It is shown that if the norm of certain matrices derived from the Jacobians of the system equations is less than one, then the WR solution converges. It is also the first time that a convergence condition has been obtained for this general kind of nonlinear system in the WR literature. Numerical experiments are provided to confirm the theoretical analysis.